Question

given the image, which of these gives enough information to use one of the theorem converses to prove that b || n? m∠7 = 68°. m∠6 = 112°, m∠4 = 68°, m∠5 = 112°, m∠2 = 112°

Explanation:

Step1: Recall angle - relationship theorems

When two lines are parallel, corresponding angles are equal, alternate - interior angles are equal, and same - side interior angles are supplementary.

Step2: Analyze the given angle $m\angle7 = 68^{\circ}$

$\angle7$ and $\angle2$ are same - side interior angles. If $b\parallel n$, then $\angle7+\angle2 = 180^{\circ}$ (same - side interior angles are supplementary). Given $m\angle7 = 68^{\circ}$, then $m\angle2=180 - 68=112^{\circ}$. If we know $m\angle2 = 112^{\circ}$, by the converse of the same - side interior angles postulate (if same - side interior angles are supplementary, then the lines are parallel), we can prove that $b\parallel n$.

Step3: Check other options

• Option A: $m\angle6 = 112^{\circ}$. $\angle6$ and $\angle7$ are vertical angles. While $m\angle6 = m\angle7=68^{\circ}$ (vertical angles are equal), this information alone does not prove $b\parallel n$.
• Option B: $m\angle4 = 68^{\circ}$. $\angle4$ and $\angle7$ are alternate exterior angles. But just knowing the measure of $\angle4$ and $\angle7$ equal does not use the converse of the theorems in the most straightforward way for this problem.
• Option C: $m\angle5 = 112^{\circ}$. $\angle5$ and $\angle7$ are corresponding angles. While if $b\parallel n$ they are equal, this does not use the converse in the best way as compared to the relationship between $\angle7$ and $\angle2$.

Answer:

D. $m\angle2 = 112^{\circ}$